The relationship between frequency and impedance in a capacitor is governed by the capacitive reactance (��Xc), which is a measure of the opposition a capacitor presents to the flow of alternating current. The formula for capacitive reactance is given by:

��=12���Xc=2πfC1

where:

- ��Xc is the capacitive reactance,
- �π is a mathematical constant (approximately 3.14159),
- �f is the frequency of the alternating current, and
- �C is the capacitance of the capacitor.

From the formula, it’s evident that capacitive reactance is inversely proportional to both frequency and capacitance. Therefore, as the frequency increases, the capacitive reactance decreases.

To understand why a higher frequency results in lower impedance for a capacitor, consider the behavior of a capacitor in an AC circuit. In an AC circuit, the voltage across a capacitor leads the current, and the relationship between voltage and current is given by:

�=�����I=CdtdV

Here:

- �I is the AC current,
- �C is the capacitance, and
- ����dtdV is the rate of change of voltage with respect to time.

At higher frequencies, the rate of change of voltage (����dtdV) becomes more rapid. As a result, the capacitor cannot fully charge or discharge during each cycle of the alternating current. This limitation means that the capacitor impedes the flow of current less effectively compared to lower frequencies.

In summary, the capacitive reactance decreases with increasing frequency because at higher frequencies, the capacitor cannot respond quickly enough to the rapidly changing voltage, resulting in lower impedance. This behavior is fundamental to the operation of capacitors in AC circuits and is crucial in applications such as filtering and coupling in electronic circuits.